課程名稱︰物理化學二 課程性質︰必修 課程教師︰鄭原忠 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2010/4/24 考試時限（分鐘）：120 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Refer to the last page for formulas of frequently used intergrals. 1.Consider a molecule rotating in a two-dimensional space described by the Hamiltonian H=L^2/2m, where Lz=-ihd/2pidφ and φ is the angular orientation (a)(5%)Show that φ+=A+exp(imφ) and φ-=A-exp(-imφ) are the two independent solutions to the time-independent Schrodinger equation. Determine the normalization constants A+ and A-. (b)(5%)Is it possible to simultaneously measure the angular orientation and angular momentum of rotating in a two-dimensional space? Explain your answer by evaluationg the commutator {φ, Lz}. 2.Use the following commutators for angular momentum operators to answer this problem:{Lx,Ly}=ihLz/2pi, {Ly,Lz}=ihLx/2pi, {Lz, Lx}=ihLy/2pi. (a)(5%)Calculate the commutators {L^2y, Lx} and {L^2z, Lx}. (b)(5%)Use the results to show that {L^2, Lx}=0. 3.The Schrodinger equation ofr rotation in three dimensions is -h^2/8pi^2I{1/sin(x)(d/dx)(sin(x)(d/dx)+1/sin^2(x)(d^2/dφ^2}Y(x,φ)= EY(x.φ). (a)(5%)Show that the rotational wavefunction φ(x,φ)=(5/16pu)^1/2* (3cos^2x-1) is an eigenfunction of the Hamiltonian. What is the energy eigenvalue? (b)(5%)Is φ(x.φ) an eigenfunction of Lz? If the answer is yes, give the l and m quantum numbers for the spherical harmonic function Y(x,φ)= φ(x,φ). 4. (10%)The 2s radial function for a hydrogen-like atom is R2s(r)=1/1.414(z/a)^3/2(1-Zr/2a)exp(-Zr/2a) Calculate <r> for an electron in the 2s orbital. 5. On the right are figures of four hydrogen atomic orbitals including a contour plot in the x-y plane, the radial function, and the radial probability distribution function. (a)(5%)Give the principle quantum number and angular momentum quantum number for each of the orbitals. Use the s,p,d,f...symbols in your answer. (b)(5%)Sort the orbitals from low energy to high energy. (c)(5%)List all allowed optical absorptions for electronic transitions between these levels. Given that the Rydberg constant is 13.6 eV, determine the transition energy for each of the allowed optical absorptions. 4.Evaluate these intergrals without explicitly carrying out the integration. Explain your results. (a)(3%)<2p1|Lz|2px>. (b)(3%)<2p1|H|3px>. (c)(3%)<2po|L^2|2po>. (d)(3%)<2po|Lz|3px>. 5.(8%)Consider the harmonic oscillator Hamiltonian:H=-h^2d^2/8pi^2mdx^2+ (1/2)mw^2x^2. Given a gaussian trial wavefunction φ(x)=Nexp(-ax^2) where N is an normalization constant and a is a positive variational parameter. Determine the variational ground-state energy using the variational theorem. Explain what you find. 6.(5%)For rotational spectroscopy the symbol J is often used to denote the angular momentum quantum number instead of l. The figure on the right is the energy levels for rigid rotor, Ej=hcBJ(J+1), where B is the rotational constant. The allowed transitions are shown as red vertical bars. Explain the selection rule, and then list the three allowed rotational transitions with lowest transition energies. | | |______________110hcB | | | |______________72hcB | | |______________42hcB energy | |______________20hcB |______________6hcB 7.Answer true or false for the following statements(3 points each): (a)Y(0,0) is and eigenfunction of L^2, Lx, Ly, and Lz. (b)The L^2 eigenvalues are degenerate except for l=0. (c)Omitting the effects of electron spin, the hydrogen n=2 shell splits into four energy levels when an external magnetic firlds is applied. (d)For a hydrogen atom in the 2px state. the possible outcomes of a measurement of Lz are -h,0 and h/2pi. (e)The magnitude of the auqular momentum of a hydrogen-like atom is proportional to l(l+1). 8.Bonus question(10 points): On the right is a rotational spectrum og a diatomic molecule measured through absorption of microwave radiation at room temperature. Explain the appearance of the spectrum including the energy spacing between the peaks and why the lowest energy transition does not have the strongest intensity. What do you expect to see when the temperature is lowered? | | | | l l | l l | l l l l absorbance | l l l l | l l l l | l l l l l | l l l l l l | l l l l l l l | l l l l l l l l |l l l l l l l l l l |___________________ energy Frequently used integral formulas:(skip) --

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